It is worth noting that in one dimension simplicial interpolation based on the Delaunay triangulation is equivalent to 1-NN for classification.
Yet, when the dimension \(d\)  is high enough, simplicial interpolation is nearly optimal both for classification and regression. Specifically, it is was shown in Theorem 3.4 in [1]} (Theorem 3.4) that simplicial interpolation benefits from a blessing of dimensionality. For large \(d\) , the excess risk of \(f_{\rm simp}\)  decreases with dimension:
\({\mathcal {R}}(f_{\rm simp}) - {\mathcal {R}}(f^*) = O\left(\frac{1}{\sqrt{d}} \right).\) 
